Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135848 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,17,12,19,30,18)(7,20,24,8,25,28,23,29)(9,16,31,15,32,36,10,33)(13,22,34,14,26,35,27,21);(1,2)(3,4)(5,7,10,22,11,24,9,14,12,25,31,35,30,23,32,21)(6,13,16,8,18,27,33,20,19,26,36,29,17,34,15,28);(2,4)(5,8,9,26,30,28,10,27,12,29,32,13,11,20,31,34)(6,14,15,24,17,22,36,7,19,21,33,23,18,35,16,25)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321664, #72321689, #72321698, … |
| maximal normal subgroup of | #72321664, #72321689, #72321698, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56193);